User:BudjarnLambeth/Sandbox2: Difference between revisions

{{main|23edo and octave stretching}}

23edo is not typically taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention.

However, when using a slightly [[stretched tuning|stretched octave]] of around 1216 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well.

Stretched 23edo is one of the best tunings to use for exploring the [[antidiatonic]] scale since its fifth is more [[consonant]] and less "[[Wolf interval|wolfish]]" than fifths in other [[pelogic family]] temperaments.

What follows is a comparison of stretched- and compressed-octave 23edo tunings.

; [[zpi|86zpi]]  

* Step size: 51.653{{c}}, octave size: 1188.0{{c}}

Compressing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this.

{{Harmonics in cet|51.653|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}

{{Harmonics in cet|51.653|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}

; [[60ed6]]

* Step size: 51.700{{c}}, octave size: 1189.1{{c}}

Compressing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 60ed6 does this. So does the tuning [[equal tuning|105ed23]] whose octave is identical within 0.01{{c}}.

{{Harmonics in equal|60|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}

{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}

; [[zpi|85zpi]]  

* Step size: 52.114{{c}}, octave size: 1198.6{{c}}

Compressing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 85zpi does this. So does the tuning [[ed9|73ed9]] whose octave is identical within 0.02{{c}}.

{{Harmonics in cet|52.114|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}

{{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}

; 23edo

* Step size: NNN{{c}}, octave size: 1200.0{{c}}  

Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.

{{Harmonics in equal|23|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}}

{{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}}

; [[WE|23et, 13-limit WE tuning]]  

* Step size: 52.237{{c}}, octave size: 1201.5{{c}}

Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.

{{Harmonics in cet|52.237|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}

{{Harmonics in cet|52.237|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}

; [[WE|23et, 2.3.5.13 WE tuning]]  

* Step size: 52.447{{c}}, octave size: 1206.3{{c}}

Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. So does the tuning [[ed10|76ed10]] whose octave is identical within 0.01{{c}}.

 

 

* Step size: 52.615{{c}}, octave size: 1210.1{{c}}

Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this.

{{Harmonics in cet|52.615|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}

{{Harmonics in cet|52.615|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}

 

 

{{Harmonics in equal|84|13|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}